Let $M$ and $N$ be two finitely generated graded modules over a standard graded Noetherian ring $R=bigoplus_{ngeq 0} R_n$. In this paper we show that if $R_{0}$ is semi-local of dimension $leq 2$ then, the set $hbox{Ass}_{R_{0}}Big(H^{i}_{R_{+}}(M,N)_{n}Big)$ is asymptotically stable for $nrightarrow -infty$ in some special cases. Also, we study the torsion-freeness of graded generalized local ...

is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E∗, particularly p–complete theories with an associated formal group of height n. We briefly remind our readers of the terms in this last sentence. A multiplicative cohomology theory E∗ is complex oriented if there exists a class x ∈ E2(...

The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes the Patterson conjecture. We also extend the range of zeta functions in con...

There exists a p-local spectrum T (m) with BP∗(T (m))= BP∗[t1, . . . , tm]. Its Adams-Novikov E2-term is isomorphic to ExtΓ(m+1)(BP∗, BP∗), where Γ(m+ 1) = BP∗(BP )/ (t1, . . . , tm) = BP∗[tm+1, tm+2, . . . ]. In this paper we determine the groups ExtΓ(m+1)(BP∗, v −1 n BP∗/In) for all m,n > 0. Its rank ranges from n + 1 to n2 depending on the value of m.

Steenrod operations were defined by Voedvodsky in motivic cohomology in order to prove the Milnor and Bloch-Kato conjectures. These operations have also been constructed by Brosnan for Chow rings [5]. The purpose of this paper is to provide a setting for the construction of the Steenrod operations in algebraic geometry, for generalized cohomology theories whose formal group law has order two. W...

This is a course in algebraic topology for anyone who has seen the fundamental group and homology. Traditionally (20 years ago), the syllabus included cohomology, the universal coefficient theorem, products, Tor and Ext and duality. We will cover all those topics, while also building a modern framework of derived categories and derived functors using the Cartan–Eilenberg method. We will apply t...

We extend the short presentation due to [Borel ’53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing and giving a new characterization of [Fulton ’92]’s definition for permutatio...

Differentials are introduced to the method of generating functions for generalized power series with exponents in a totally ordered Abelian group. Logarithmic analogue of cohomology residues is defined to equate coefficients.

DI(4) is the only known example of an exotic 2-compact group, and is conjectured to be the only one. In this work, we study generalized cohomology theories for DI(4) and its classifying space. Specifically, we compute the Morava K-theories, and the P (n)-cohomology of DI(4). We use the non-commutativity of the spectrum P (n) at p = 2 to prove the non-homotopy nilpotency ofDI(4). Concerning the ...